Feferman awarded Rolf
Schock Prize in logic and philosophy

Solomon Feferman, the Patrick Suppes Family Professor in the
School of Humanities and Sciences at Stanford, was awarded Sweden's
Rolf Schock Prize in logic and philosophy on May 15. The Schock
prizes are awarded every two years in the fields of logic and
philosophy, mathematics, the visual arts and music. The prize
citation lauded Feferman, who has made important contributions in
key areas of logic, "for his works on the arithmetization of
metamathematics, transfinite progressions of theories and
predicativity."

Solomon
Feferman is the recipient of the Rolf Schock Prize for his
accomplishments in logic and philosophy. Photo: Anita Feferman

"Though the Schock Prize is of relatively recent vintage, it
is a major honor; indeed, perhaps the highest prize currently
awarded for philosophy," said Kenneth A. Taylor, professor and
chair of the Department of Philosophy. "The [short] list of past
winners of the logic and philosophy prize reads like an honor roll
of the greatest philosophers and logicians of the mid to late 20th
century. They include Willard Van Orman Quine, Dana Scott, Michael
Dummett, John Rawls and Saul Kripke. Sol Feferman richly deserves
to be counted as one among the true giants of 20th-century logic
and philosophy."

Arithmetization is a method that was introduced by Austrian
logician Kurt Gödel to prove his famous incompleteness
theorems for axiomatic systems of mathematics. It is now a standard
technique in logic, and it was Feferman who first systematically
established its scope and limitations. Gödel's first
incompleteness theorem states that any consistent and sufficiently
expressive axiomatic theory cannot prove all true statements about
the whole numbers. The method of arithmetization uses an assignment
of "Gödel numbers" to sequences of symbols to translate
logical properties of a theory of numbers into its own language. In
that way, the consistency of the theory can be re-expressed as a
statement about numbers. Feferman's work on arithmetization led him
to establish a sharpened and very general formulation of
Gödel's second incompleteness theorem, which says that the
consistency of a theory -- if that theory is indeed consistent --
cannot be proved within the theory itself.

"In
a progression of theories based on an initial theory of numbers,
one attempts to overcome incompleteness by successively adding
consistency statements or stronger principles reflecting on the
systems in question," Feferman explained in an e-mail interview.
"Not only can this procedure be repeated any finite number of
times, it can also be iterated into the transfinite." Feferman
investigated how far into the transfinite such progressions could
be carried, and established fundamental completeness and
incompleteness results for them, depending on the way that they are
generated; these results made essential use of his work on
arithmetization.

Feferman then turned to the study of predicativity, in which
certain definitions ubiquitous in mathematics, such as that for the
least upper bound of a set of real numbers, are restricted to avoid
apparent circularities. Using a formulation of this idea in terms
of a transfinite progression of theories, Feferman determined the
exact limit of predicativity as measured by a certain transfinite
ordinal number. In later work he greatly enlarged the understanding
of the scope of predicativity, in part by demonstrating that
practically all the mathematical principles needed for scientific
applications can be carried out entirely according to predicative
restrictions.

Feferman obtained his doctorate in mathematics from the
University of California-Berkeley in 1957. Since 1956, he has
taught at Stanford in the departments of Mathematics and
Philosophy. He served as chair of the Mathematics Department from
1985 to 1992 and as president of the Association for Symbolic Logic
from 1980 to 1982. He is a fellow of the American Academy of Arts
and Sciences and editor-in-chief of Kurt Gödel's Collected
Works, Volumes I-V. He and his wife, Anita Burdman Feferman,
are currently working on a biography of the logician Alfred Tarski,
who was Feferman's doctoral adviser at UC-Berkeley.